The formal strong completeness of partial monoidal Boolean BI
Journal of Logic and Computation
This paper presents a self-contained proof of the strong completeness of the labeled tableaux method for partial monoidal Boolean BI: if a formula has no tableau proof then there exists a counter-model for it which is simple. Simple counter-models are those which are generated from the specific constraints that occur during the tableaux proof-search process. As a companion to this paper, we provide a complete formalisation of this result in Coq 1 and discuss some of its implementation details.
... n example, Roy Dyckhoff and Sara Negri used a labeled sequent calculus to propose decision methods for Gödel-Dummett logic  . The Logic of Bunched Implications [29, 30] called BI is a sub-structural logic usually considered as the foundation of separation logics [19, 26] and spatial logics  . It contains both additive operators like ∧, ∨ and → and multiplicative operators like * and − * . The multiplicative operators are those of multiplicative intuitionistic Linear Logic  . The additives can be interpreted either as in intuitionistic logic which gives rise to intuitionistic BI , or as Boolean operators which gives rise to Boolean BI [30, 13] . Classical BI [2, 3] is another variant of BI which combines Boolean additives with classical multiplicatives. The core semantic link between BI and separation logic can be summed up in the Kripke sharing interpretation of the multiplicative conjunction: m and a A and b B The ternary composition/decomposition relation − • − − reads either as "m is a result of the composition of a and b" or "m can be decomposed into a and b." The interpretation of composition/decomposition depends on the variant of the logic and/or model, see [22, 24, 5] for an overview of some possible interpretations of this relation. In this paper we will focus on Boolean BI (denoted BBI), more precisely on partial monoidal BBI. In this case, the composition • is a partial monoidal operator and the relation is the identity or at least a congruence relation w.r.t. the partial monoidal composition •. This is not a restriction since all the models of separation logic and abstract separation logic [7, 5, 21] are in fact partial monoids  . Contrary to what happened with intuitionistic BI which was well defined by a cut-free bunched sequent calculus since its inception  , later completed with a decidability result  , the proof theory of BBI was, at first, not very well understood. In  , it is defined as the addition of a double negation principle/axiom to intuitionistic BI, but of course, with this axiom, you loose either cut-elimination or the bunched sequent calculus. In , a sound and complete Hilbert style proof system is given for a variant of BBI called relational BBI or non-deterministic BBI. Later,  provided a cut-free Display-style sequent calculus for relational BBI. In , a sound labeled tableaux calculus is given for partial monoidal BBI, leading to an embedding of intuitionistic BI into Boolean BI, a result which was quite unexpected at that time. 4 But it was still unknown whether relational and partial monoidal BBI coincide or not, or whether BBI was decidable or not. Then, the situation improved a lot with a model that distinguishes relational and partial monoidal BBI  as well as other variants of Boolean BI, leading to a family of different logics  , and an undecidability result obtained for the whole family of BBI/separation logics, independently and simultaneously in  and  . We also point out the undecidability result for Classical BI [4, 20] . These undecidability results doomed the different attempts made at providing a decision procedure for BBI either through Display logic  or through tableaux calculi  . However in this paper, we explain how the labeled tableaux calculus can still be useful as a tool for the study of the properties of BBI, like finer completeness results. We believe that the labeled calculus can also serve as an effective semi-decision algorithm for partial monoidal BBI, but we will only discuss this as a perspective. This work also comes as a complement to  , the knowledge of which being advised but not required. Let us give a quick overlook of the content of the upcoming sections: • in Section 2, we describe a framework of labels represented by words and constraints between those labels that can be used as a syntactic representation for partial monoids. The solutions of those constraints, partial monoidal equivalences, give a foundation to the semantics of (partial monoidal) BBI; • in Section 3, we introduce the syntax and Kripke semantics of (partial monoidal) BBI and the notion of (counter-)model; 4 and it is the completeness of a nearly isomorphic labeled tableau calculus that we establish here. tutions such that ρ(a) = ρ (a) for every letter a ∈ A C . Then (X, C) is satisfied in (K, ∼, , ρ) if and only if it is satisfied in (K, ∼, , ρ ). Proof. Left to the reader. Definition 38 (Realizability). A BBI-tableau is realizable if at least one of its branches has a model. Proposition 39. Closed BBI-tableaux are not realizable. Proof. We prove that a closed branch (X, C) cannot be satisfied in any (K, ∼, , ρ). Let us suppose the contrary and proceed by case analysis on the closure condition: